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Properties

Label 118.2.c.a
Level $118$
Weight $2$
Character orbit 118.c
Analytic conductor $0.942$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [118,2,Mod(3,118)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(118, base_ring=CyclotomicField(58))
 
chi = DirichletCharacter(H, H._module([50]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("118.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 118 = 2 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 118.c (of order \(29\), degree \(28\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.942234743851\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(2\) over \(\Q(\zeta_{29})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{29}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 2 q^{2} - q^{3} - 2 q^{4} + q^{5} + q^{6} + 4 q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 2 q^{2} - q^{3} - 2 q^{4} + q^{5} + q^{6} + 4 q^{7} + 2 q^{8} + q^{9} - q^{10} - 28 q^{11} - q^{12} - q^{13} - 33 q^{14} - 7 q^{15} - 2 q^{16} + 3 q^{17} - q^{18} + 5 q^{19} + q^{20} - 24 q^{21} - q^{22} - 8 q^{23} + q^{24} - 3 q^{25} + q^{26} - 88 q^{27} + 4 q^{28} + 5 q^{29} + 7 q^{30} - 6 q^{31} + 2 q^{32} - 4 q^{33} - 32 q^{34} + 3 q^{35} + q^{36} - 45 q^{37} - 5 q^{38} - 8 q^{39} - 30 q^{40} - 12 q^{41} - 5 q^{42} + 15 q^{43} + q^{44} + 21 q^{45} + 37 q^{46} - 112 q^{47} - q^{48} + 62 q^{49} + 3 q^{50} + 116 q^{51} + 28 q^{52} - q^{53} + 88 q^{54} + 79 q^{55} - 4 q^{56} - 10 q^{57} + 24 q^{58} + 58 q^{59} + 22 q^{60} + 60 q^{61} + 35 q^{62} - 57 q^{63} - 2 q^{64} + 75 q^{65} + 91 q^{66} - 14 q^{67} + 32 q^{68} + 100 q^{69} - 90 q^{70} + 69 q^{71} - q^{72} - 51 q^{73} + 16 q^{74} + 35 q^{75} + 5 q^{76} - 57 q^{77} + 8 q^{78} - 54 q^{79} + q^{80} + 10 q^{81} + 12 q^{82} - 91 q^{83} + 5 q^{84} - 4 q^{85} - 15 q^{86} + 7 q^{87} - q^{88} - 56 q^{89} - 79 q^{90} + 7 q^{91} - 8 q^{92} - 156 q^{93} - 4 q^{94} + 25 q^{95} + q^{96} - 14 q^{97} - 4 q^{98} - 92 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.907575 + 0.419889i −2.70071 − 0.909974i 0.647386 − 0.762162i 1.33357 + 0.802381i 2.83318 − 0.308127i −0.202759 + 3.73968i −0.267528 + 0.963550i 4.07748 + 3.09962i −1.54722 − 0.168271i
3.2 −0.907575 + 0.419889i 1.79736 + 0.605602i 0.647386 − 0.762162i 0.860217 + 0.517576i −1.88553 + 0.205064i 0.0637765 − 1.17629i −0.267528 + 0.963550i 0.475483 + 0.361452i −0.998036 − 0.108543i
5.1 0.947653 + 0.319302i 0.223080 + 0.134223i 0.796093 + 0.605174i −0.887916 + 2.22850i 0.168545 + 0.198426i 1.25178 − 0.579135i 0.561187 + 0.827689i −1.37348 − 2.59065i −1.55300 + 1.82833i
5.2 0.947653 + 0.319302i 0.911025 + 0.548146i 0.796093 + 0.605174i 1.20694 − 3.02920i 0.688312 + 0.810344i −4.32641 + 2.00161i 0.561187 + 0.827689i −0.875722 − 1.65179i 2.11099 − 2.48525i
7.1 0.561187 + 0.827689i −0.0645121 + 1.18986i −0.370138 + 0.928977i −1.67396 + 0.774454i −1.02103 + 0.614336i −0.350620 + 1.26282i −0.976621 + 0.214970i 1.57082 + 0.170837i −1.58041 − 0.950901i
7.2 0.561187 + 0.827689i 0.0610915 − 1.12677i −0.370138 + 0.928977i 1.72581 − 0.798442i 0.966895 − 0.581762i 0.0672491 − 0.242209i −0.976621 + 0.214970i 1.71654 + 0.186685i 1.62936 + 0.980354i
9.1 −0.647386 + 0.762162i −1.24224 − 0.944324i −0.161782 − 0.986827i 0.544974 + 1.02793i 1.52393 − 0.335443i 4.79708 + 0.521714i 0.856857 + 0.515554i −0.151181 − 0.544504i −1.13626 − 0.250109i
9.2 −0.647386 + 0.762162i 1.89117 + 1.43763i −0.161782 − 0.986827i −0.0313387 − 0.0591111i −2.32003 + 0.510677i 1.86365 + 0.202684i 0.856857 + 0.515554i 0.707163 + 2.54697i 0.0653405 + 0.0143825i
15.1 0.994138 − 0.108119i −1.48752 − 1.75125i 0.976621 − 0.214970i 0.597385 − 0.454120i −1.66815 − 1.58015i −0.687647 − 1.72586i 0.947653 − 0.319302i −0.368797 + 2.24956i 0.544784 − 0.516047i
15.2 0.994138 − 0.108119i 0.910235 + 1.07161i 0.976621 − 0.214970i −1.69766 + 1.29053i 1.02076 + 0.966916i −0.192203 − 0.482393i 0.947653 − 0.319302i 0.165523 − 1.00964i −1.54818 + 1.46651i
17.1 0.561187 − 0.827689i −0.0645121 − 1.18986i −0.370138 − 0.928977i −1.67396 − 0.774454i −1.02103 − 0.614336i −0.350620 − 1.26282i −0.976621 − 0.214970i 1.57082 − 0.170837i −1.58041 + 0.950901i
17.2 0.561187 − 0.827689i 0.0610915 + 1.12677i −0.370138 − 0.928977i 1.72581 + 0.798442i 0.966895 + 0.581762i 0.0672491 + 0.242209i −0.976621 − 0.214970i 1.71654 − 0.186685i 1.62936 − 0.980354i
19.1 −0.468408 + 0.883512i −0.471025 + 0.446178i −0.561187 − 0.827689i −4.10385 + 0.903326i −0.173572 − 0.625149i −1.33582 + 1.01547i 0.994138 − 0.108119i −0.139628 + 2.57528i 1.12418 − 4.04893i
19.2 −0.468408 + 0.883512i 2.44117 − 2.31240i −0.561187 − 0.827689i −0.278726 + 0.0613523i 0.899567 + 3.23995i −1.68570 + 1.28143i 0.994138 − 0.108119i 0.449704 − 8.29431i 0.0763523 − 0.274996i
21.1 0.856857 + 0.515554i −0.545651 + 1.36948i 0.468408 + 0.883512i 1.06298 + 0.115606i −1.17359 + 0.892136i −1.84232 − 0.620752i −0.0541389 + 0.998533i 0.600246 + 0.568583i 0.851223 + 0.647083i
21.2 0.856857 + 0.515554i 0.717744 − 1.80140i 0.468408 + 0.883512i −1.63137 − 0.177423i 1.54372 − 1.17351i 2.39547 + 0.807128i −0.0541389 + 0.998533i −0.551904 − 0.522792i −1.30638 − 0.993087i
25.1 −0.796093 − 0.605174i −0.968824 − 1.82740i 0.267528 + 0.963550i −1.89443 − 1.79450i −0.334619 + 2.04109i −2.16221 + 2.54555i 0.370138 − 0.928977i −0.717196 + 1.05779i 0.422159 + 2.57505i
25.2 −0.796093 − 0.605174i −0.387362 − 0.730642i 0.267528 + 0.963550i 2.24264 + 2.12434i −0.133790 + 0.816080i 1.31065 − 1.54301i 0.370138 − 0.928977i 1.29977 − 1.91702i −0.499753 − 3.04836i
27.1 −0.267528 + 0.963550i −1.03285 − 1.52333i −0.856857 − 0.515554i 0.0803191 − 1.48140i 1.74412 − 0.587663i 0.314131 − 1.91611i 0.725995 − 0.687699i −0.143359 + 0.359803i 1.40591 + 0.473708i
27.2 −0.267528 + 0.963550i 0.700518 + 1.03319i −0.856857 − 0.515554i −0.0468430 + 0.863969i −1.18294 + 0.398577i −0.504289 + 3.07603i 0.725995 − 0.687699i 0.533665 − 1.33940i −0.819945 − 0.276272i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.c even 29 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 118.2.c.a ✓ 56
4.b odd 2 1 944.2.m.a 56
59.c even 29 1 inner 118.2.c.a ✓ 56
59.c even 29 1 6962.2.a.bi 28
59.d odd 58 1 6962.2.a.bj 28
236.h odd 58 1 944.2.m.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
118.2.c.a ✓ 56 1.a even 1 1 trivial
118.2.c.a ✓ 56 59.c even 29 1 inner
944.2.m.a 56 4.b odd 2 1
944.2.m.a 56 236.h odd 58 1
6962.2.a.bi 28 59.c even 29 1
6962.2.a.bj 28 59.d odd 58 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{56} + T_{3}^{55} + 3 T_{3}^{54} + 34 T_{3}^{53} + 40 T_{3}^{52} + 137 T_{3}^{51} + 913 T_{3}^{50} + \cdots + 863041 \) acting on \(S_{2}^{\mathrm{new}}(118, [\chi])\). Copy content Toggle raw display